Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 3 - Differentiation - 3.2 Exercises: 10

Answer

Rolle's theorem can be applied; $c=4.$

Work Step by Step

Since $f(x)$ is a polynomial, it is continuous for all values of $x$ and differentiable at every value of $x$. $f(2)=f(6)=-7.$ Since $f(x)$ is continuous over $[2 , 6]$ and differentiable over $(2, 6)$, applying Rolle's Theorem over the interval $[2, 6]$ guarantees the existence of at least one value $c$ such that $2\lt c\lt6 $ and $f'(c)=0.$ $f'(x)=2x-8\to f'(x)=0\to 2x-8=0\to c=4.$
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